Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements (Q380497)

The authors consider a quadratic elliptic control problem with pointwise control constraints. As state equation a second order elliptic differential equation with Dirichlet boundary conditions in a polygonal domain \(\Omega \subset \mathcal{R}^2\) is given. For the approximation of the state equation as well as the co-state variables Raviart-Thomas mixed finite elements of order \(k=1\) and for the control piecewise linear not necessarily continuous functions are used. By following an idea of \textit{C. Meyer} and \textit{A. Rösch} [SIAM J. Control Optim. 43, No. 3, 970--985 (2004; Zbl 1071.49023)] superconvergence of order \(\mathcal{O}(h^2)\) is achieved by postprocessing the discretized adjoint state. A numerical experiment is provided.